[17] Hadamard’s formula and variation of domain-functions

Abstract

In this important paper, Schiffer opens new pathways for the variational method. He had previously introduced a generalization of the classical formula of Hadamard (Mémoires présentés par divers savants à l’Académie de Sciences, 33(4), 1–128, 1908) for variation of Green’s function. Hadamard’s analysis relied on normal displacement of the boundary and required that the domain be smoothly bounded. Recognizing the need for greater generality for application to extremal problems, Schiffer (Amer. J. Math., 65, 341–360, 1943) developed the method of interior variation and applied it to the coefficient problem for univalent functions. In Schiffer (Amer. J. Math., 68, 417–448, 1946), he turns attention to a variety of extremal problems involving quantities expressible in terms of Green’s function: transfinite diameter, harmonic measures, canonical mappings of multiply connected domains, the Bergman kernel function, etc. Here the insight is that a variation of Green’s function induces a variational formula for any such quantity.